Towards Lax Formulation of Integrable Hierarchies of Topological Type

Johannes van de Leur, Guido Carlet, Sergey Shadrin, Hessel Posthuma

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

To each partition function of cohomological field theory one can associate
an Hamiltonian integrable hierarchy of topological type. The Givental group acts on
such partition functions and consequently on the associated integrable hierarchies. We
consider theHirota and Lax formulations of the deformation of the hierarchy of N copies
of KdV obtained by an infinitesimal action of the Givental group. By first deforming
the Hirota quadratic equations and then applying a fundamental lemma to express it in
terms of pseudo-differential operators, we show that such deformed hierarchy admits an
explicit Lax formulation. We then compare the deformed Hamiltonians obtained from
the Lax equations with the analogous formulas obtained in Buryak et al. (J Differ Geom
92:153–185, 2012), Buryak et al. (J Geom Phys 62:1639–1651, 2012) to find that they
agree.We finally comment on the possibility of extending theHirota and Lax formulation
on the whole orbit of the Givental group action.
Original languageEnglish
Pages (from-to)815-849
Number of pages35
JournalCommunications in Mathematical Physics
Volume326
DOIs
Publication statusPublished - 2014

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